International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
411
Type-2 Fuzzy Cerebellar Model Articulation Controller-Based Learning
Rate Adjustment for Blind Source Separation
Meng-Tzu Huang, Ching-Hung Lee, and Chin-Min Lin
Abstract1
can be solved by independent component analysis (ICA)
which measures the non-Gaussian signals to make the
Blind source separation (BSS) is a technique for signals independent to each other [14, 15, 20, 23]. There
recovering a set of source signals without a priori in- are some typical adaptive algorithms of ICA such as
formation on the transformation matrix or the prob- natural gradient algorithm [1, 45], the equivariant adapability distributions of source signals. Based on sepa- tive separation via independence (EASI) [11, 14]. For
ration results of outputs, this paper proposes the in- the gradient algorithm, the choice of learning rate (or
terval type-2 fuzzy cerebellar model articulation con- step-size) reflects a tradeoff between mis-adjustment and
troller (T2FCMAC)-based learning rate adjustment the speed of convergence. When a fixed large learning
for the BSS. The adopted T2FCMAC system has the rate is chose, we have a fast initial convergence and reability of generating the proper learning rate by us- sult a larger steady state; and the slow convergence will
ing the inputs of second- and higher order correlation occurs when a small learning rate is set.
coefficients of output components. In addition, to enMany approaches have been proposed for choosing
hance the performance of the T2FCMAC-based learning rate on gradient algorithm. The non-adaptive
learning rate approach, the T2FCMAC system is op- leaning rate selection, called gear-shifting or annealing
timized by particle swarm optimization (PSO) algo- rule [14, 43, 44], starts from a large value and decreases
rithm by the performance index of second-order cor- to zero. But it may be unstable and inefficient. On the
relation measure. Simulation and comparison results other hand, the adaptive methods have been proposed to
are introduced to show the effectiveness and per- find the suitable learning rates [4, 13, 17, 19, 36]. Howformance of the proposed approach.
ever, it should note that these methods are based on the
separation states of the BSS. If the separated signals of
Keywords: Blind source separation, independent com- output are worse, the learning rate should be increase to
ponent analysis, cerebellar model articulation control- speed up the separation; if the separated signals are well,
ler, interval type-2 fuzzy system, particle swarm opti- the learning rate should be decrease (or small) to diminmization.
ish the mis-adjustment of convergence. In literature [19,
35], a fuzzy-based learning rate determination was pro1. Introduction
posed for adaptive BSS. The fuzzy-based method performs good signal separation but the membership funcIn recent years, blind source separation (BSS) has re- tions and the fuzzy rule should be set up and need exceived attentions in several areas, such as communica- pert’s experience.
tions and speech processing, various biomedical signal
In order to make the learning rate selection more
processing, and image processing [3-5, 7-8, 10-11, 13-15, flexible and perform better, we adopt a novel interval
39-41, 43, 44]. It has become an active research area in type-2 fuzzy cerebellar model articulation controller
both statistical signal processing and unsupervised neural (T2FCMAC) to adjust the learning rate for the BSS. The
learning. The purpose of the BSS is to recover source cerebellar model articulation controller (CMAC) models
signals from the observed mixed signals. The problem the structure and functions of the part of the brain known
as cerebellum proposed by Albus [1, 2]. It is a class of
Corresponding Author: Ching-Hung Lee is with the Department of neural network and has been adopted to solve the probMechanical Engineering, National Chung Hsing University, 250 lems in many fields since its fast learning property, good
Kau-Kung Rd., Taichung, Taiwan, 402.
generalization capability, and ease to implement [1, 2, 22,
E-mail: [email protected]
Meng-Tzu Huang is with Department of Electrical Engineering, Yuan 33, 34, 37]. Recently, the concept of fuzzy is considered
to combine with CMAC [22, 32, 33] and type-2 fuzzy
Ze University, Taoyuan, Taiwan.
Chih-Min Lin is with the Department of Electrical Engineering and CMAC has been proposed [20, 27]. A novel interval
Innovation Center for Big Data and Digital Convergence, Yuan Ze type-2 fuzzy cerebellar model articulation controller
University, Taoyuan, Taiwan. E-mail: [email protected]
Manuscript received 4 Oct. 2013; revised 15 April 2014, 12 June 2014; (T2FCMAC) has been proposed to illustrate the performance of the interval type-2 fuzzy mechanism with
accepted 15 July, 2014.
© 2014 TFSA
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
412
CMAC combination [27]. As our previous results of [27],
the T2FCMAC is a more generalized network with better
learning ability and has lower computational complexity
for practical implementation. Therefore, according to the
previous results and the robust characteristics, the learning rate in the BSS can be selected by T2FCMAC system adaptively. Furthermore, to enhance the performance of the BSS, the particle swarm optimization (PSO)
is adopted to adjust the parameter in the T2FCMAC system [18, 21, 24, 26], called T2FCMAC_PSO. Therefore,
the proposed approach for ICA method becomes efficient, which is verified by simulation and comparison
results.
The rest of this paper is organized as follows. Section
2 introduces briefly the blind source separation problem
and EASI method. The major contribution is introduced
in Section 3 includes learning rate adjustment approach
using T2FCMAC and optimization by PSO. Section 4
introduces the simulation results to illustrate the effectiveness of our approach. Finally, conclusion is given in
Section 5.
2. Blind Source Separation
In recent years, the BSS problem has received considerable attention from the signal processing community
and the neural network community. It has become an
active research area in both statistical signal processing
and unsupervised neural learning [14, 16, 23]. The object
of the BSS is to recover the unknown source signals
from observed mixed signals.
A. Problem formulation
Herein, we assume that there are n observed signals
X(t)=[x1(t), x2(t), …, xn(t)]T are collected by a set of n
sensors. The observed signals X(t) are mixed by
full-rank n×n mixing matrix A (A is nonsingular and
unknown) with n unknown source signals S(t)=[s1(t),
s2(t), …, sn(t)] T. Thus, the mixed model is defined as
(1)
Xt   ASt  .
The goal of the BSS is to recover the unknown source
signals S(t) from the observed mixed signals X(t) without knowing A. There exists a separated model defined
as
(2)
Yt   Wt Xt 
T
where Y(t)=[y1(t), y2(t), …, yn(t)] is separated output
signal, W(t) is separating matrix, and t is time instant.
According to (1) and (2), we have
(3)
Yt   Wt ASt   Bt St  .
-1
This means that there is the ideal result W(t)=A such
that B(t)=I and then we have Y(t)=S(t). Therefore, the
source signals can be separated from the mixed signals.
St 
Xt 
Yt 
 t 
BSS
Figure 1. The diagram of the BSS problem.
B. Gradient algorithm for BSS
Gradient algorithm is one of popular method for independent component analysis (ICA), it is used to update
the separating matrix W. The update law is
J W 
(4)
W t  1  W t    t 
W  W t 
W
where η(t) is a learning rate, and J(.)is the objective
function of ICA. In this paper, the EASI algorithm is
used [14, 16]
Wt  1  Wt    t [I  f Yt Y T t 
(5)
T
T
 Yt  f Yt   Yt Y t ]Wt 
where I is identity matrix, f(Y(t))=Y3(t) is a nonlinear
transformation function in array power. In order to avoid
the amplitude of the separated signals Y(t) over magnifying, we normalize each vector wi of the separating matrix W(t) as
ˆi 
w
wi
wi
(6)
where wi=[wi1, wi2, ..., win], ||．|| denotes Euclidean norm
operation. The separating matrix W will be replaced by
W  [w1T , w T2 ,..., w Tn ]T . As above, our purpose is to select
adaptive learning rate η(t) so that we have fast convergence and small mis-adjustment.
In addition, the objective function is chosen by the
feature of ICA. The mutual information is usually chosen to be the objective function to measure the independence of signals. The so-called Kullback–Leibler
divergence for mutual information is defined as [14, 16]
I W    pY; W  log
pY; W 
dY
i1 pi y i ; W 
n
(7)
In general, the mutual information is nonnegative when
the components of separated signals Y are independent
when I(W)=0.
To measure the statistical performance, the
cross-talking error is adopted to be the performance index (PI) [24, 35]
n  n
 n  n

bij
bij
(8)
 1    
 1
PI    
 i 1  j 1 max k bkj

i 1  j 1 max b
k
ik




where B(t)={bij}=W(t)A is an n×n matrix, maxk|bik| =
max{|bi1|, ..., |bin|}, and maxk|bkj| = max{|b1j|, ..., |bnj|}.
M.-T. Huang et al.: Type-2 Fuzzy Cerebellar Model Articulation Controller-Based Learning Rate Adjustment
413
separation state of yi(t) is not good, i.e., yi(t) is
correlated with another output component.
(3) The separation state of yi(t) is worse if either Di(t)
or HDi(t) is large. That is, the output component
C. Dependence measure
yi(t) is correlated strongly with the other outputs.
The mutual information I(W) is a good measure of the
The problem considered in this paper is to select the
dependence between the signals. Due to the unknown
probability density function of the separated signals Y(t), proper learning rate (t) for EASI such that the mixed
it cannot be used to evaluate the dependence between signals are separated. Since Di(t) and HDi(t) perform the
each output component at the separation stage of the dependence measure of separated signals, they present
source signals. Herein, we introduce the calculation of the information of the learning rate selection in the BSS.
dependence measure second-order correlation measure D In order to obtain the suitable learning rate i(t), both
and high order correlation measure HD. Assume that yi(t) Di(t) and HDi(t) are adopted to be the inputs of the
and yj(t) are two different components of signals Y(t). learning rate selection system. In this paper, we adopt the
These two signals can be classified with a second-order T2FCMAC to develop the learning rate selection system.
correlation measure and high order correlation measure
[21, 35]. The second-order correlation coefficient is defined as
BSS
s1 t 
x1 t 
y1 t 
Cij
covy i t , y j t 
,
rij 

Cii C jj
covy i t cov y j t 
s n t 
x n t 
y n t 
for i, j = 1, 2, ..., l, i≠j
(9)
where covxt , y t   Ext   mx y t   my  , cov[x(t)]=
When PI is close to zero, the ICA will have a good performance.

1
2
 hrij  .
n  1 i1, j i
(12)
According to the above description, the measures Di(t)
and HDi(t) can describe the dependence of the output
component yi(t) with the others. Hence, we have the basic observations as follows.
(1) The separation state of yi(t) is good if both Di(t)
and HDi(t) are sufficiently small. That is, the output
component yi(t) are almost independent to each
other.
(2) If either Di(t) or HDi(t) is not small enough, the
ηn t 
D1 t 
HD1 t 
…
HDi t   HDy i t  
η1 t 
…
where  y i t   y i2 t   y 3i t  is also a nonlinear transformation function. As (2), we have
…
2
…
…

E xt   mx  , and mx  Ext  . Note that the calculations of the covariance and mean of signal yj(t) is between 0 to t time instant, which means that mx  Ext 
provides the mean value of [x(0), …, x(t)]. According to
the correlation coefficient rij, the second-order correlation measure can be defined as
1
2
(10)
Di t   Dy i t  
 rij  .
n  1 i1, j i
It is useful to adopt Di(t) to measure the dependence between two Gaussian signals. However, a limitation of
ICA is that there is at most one Gaussian source signal.
Therefore, we consider the high order correlation measure to ensure the restriction of ICA. The high order correlation measure is defined as
HCij
cov y i t , y j t 
(11)

hrij 
H ii C jj
cov y i t cov y j t 
Dn t 
HDn t 
Figure 2. The illustration of the proposed T2FCMAC-based
adaptive learning rate adjustment method for BSS.
3. T2FCMAC-Based Learning Rate Adjustment
and Optimization
This section introduces the interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC) -based
learning rate adjustment method for the BSS. In addition,
we adopt the particle swarm optimization algorithm to
optimize the T2FCMAC to enhance the performance of
signal separation. The T2FCMAC implements the interval type-2 fuzzy logic system in CMAC structure. It can
be simplified to an interval type-2 fuzzy neural network,
a fuzzy neural network, and a fuzzy cerebellar model
articulation controller (FCMAC, or called type-1
CMAC) in some special cases [9, 12, 25, 27-29, 34, 38,
42]. Hence, this T2FCMAC is a generalized network,
has lower computation complexity for practical implementation, and has better learning ability to adjust the
learning rate for the BSS. The illustration of the proposed T2FCMAC-based learning rates adjustment for
the BSS system is shown in Fig. 2. The learning rate
(t)=diag[1(t), 2(t), …, n(t)] of the EASI algorithm is
adjusted by each T2FCMAC with correlation measure
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
414
Di(t) and HDi(t) inputs, i=1, 2, ..., n. Note that different
output components require different learning rates.
Therefore, the separated signals with higher dependence
between the components should be adjusted by larger
learning rate. On the other hand, the signals with lower
dependence between the components should be adjusted
by smaller learning rate. Herein, the T2FCMAC is optimized by the particle swam optimization algorithm to
enhance the performance of our approach.
h11i t 

i
1b
t 
w11i t 

…
HDi t 
wi b t 
…
hi b t 
…
Di t 
i t 
wni  nB t 
2i b t 
hni  nB t 
Input Space
Output Space
Weight Space
Receptive-field Space
1 b
Figure 3. Structure of the T2FCMAC.
HDi t 
~
F213
8
~
F233
~
F232
7
~
F222
~
F221
~
F241
2
~
F211 1
~
F231
1 2
~
F111
Layer 1
3
~
F121
Layer 3
4
5 6
~
F112
7
Di t 
9
~
F123
~
F132
~
F142
8
~
F113
~
F122
~
F131
~
F141
(15)
rb
4
Layer 4
Layer 2
State (5, 5)
6
~
F212 5
3
~
F242
rb
1

qr  p r  b
1
2
, p rb  qr  p rb
 rb  2
1
p
p


r b
r b
2

 r b ,
qr  p r  b
 F~ qr   
3
p r  b  qr

2
3


 rb p 3  p 2 , p rb  qr  p rb
r b
r b

0 ,
otherwise

9
~
F243
rb
where r=1, 2, q1 , q2   Di t , HDi t  ,   1, 2, ..., n , and
b  1, 2, ..., nB . The interval type-2 asymmetric triangular
membership function is defined as
 qr  pr1b
1
2
 p 2  p 1 , pr  b  q r  p r  b
r b
 r b

1 ,
qr  pr2b
(14)
 F~ qr    3
p
q

2
3

r
b
r

, prb  qr  prb
 pr3b  pr2b

0 ,
otherwise
and
rb
Fuzzification Space
~
F223
Input Space: The given control space is equally divided into nE regions (elements) in this space. The
T2FCMAC in two dimensions with nE=9 is shown in Fig.
4. The number of nE is termed as the resolution.
Fuzzification Space: This space shows the fuzzification operation of interval type-2 fuzzy systems. According to the concept of CMAC, n elements form a block
and n layer present in CMAC. The illustration example
in Fig. 4 shows four elements form a full block. Therefore, there are four layer (n=4) in the fuzzification space
of T2FCMAC and three blocks (nB =3) in each layer.
Herein, we use the interval type-2 triangular asymmetric
fuzzy membership function in each block [27-31]
(13)
rb   F~ qr   [ rb rb ]  [  F~ qr   F~ qr ]
~
F133
~
F143
Layer 1
where p1 , p 2 , p 3 , p1 , p 2 , and p 3 indicate the posi-
Layer 2
tions of three vertices for the upper and the lower asymmetric triangular membership function, respectively; 
is the magnitude of the lower membership satisfying
0.5    1 to avoid the invalid result. The asymmetric
membership functions are not only more flexible than
symmetric ones but also provide the advantage of
achieving the same performance with fewer rules [6,
27-31]. Moreover, the triangular shape makes them
lower computational effort. In order to avoid unreasonable interval type-2 fuzzy membership function, the following conditions should be constrained, p1  p 2  p 3 ,
Layer 3
Layer 4
Figure 4. T2FCMAC with nB =3, nE =9, n=4 and its organization of receptive-field space activated by state (5, 5).
A. Structure of interval type-2 fuzzy CMAC
Herein, we introduce the interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC), which
is fed by D and HD to generate the proper learning rate
for EASI algorithm. The network structure of
T2FCMAC is shown in Fig. 3. The T2FCMAC has two
inputs, Di(t) and HDi(t), and one output, i(t). There are
five spaces in the T2FCMAC: Input Space, Fuzzification
Space, Receptive-field Space, Weight Space, and Output
space. The signal propagation and operation functions of
each space are described as follows.
p  p  p , p 1  p , p 3  p , and p1    p 2  p1  
1
2
3
1
3
p  p3   p3  p2  .
2
Receptive-field Space: Each location of fuzzification
space corresponds to an area in this space. By using
t-norm product, the receptive-field space is defined as
M.-T. Huang et al.: Type-2 Fuzzy Cerebellar Model Articulation Controller-Based Learning Rate Adjustment
2
2
hb ]  [ rb
hb  [h b

r 1
r b
].
(16)
r 1
Note that the outputs of fuzzification space are also interval sets, the outputs of receptive-field space are interval sets as well. Accordingly, the output of this space
consists of its lower bound hb and upper bound hb .
Weight Space: Each location of receptive-field space
corresponds to a particular adjustable value in the weight
space. The weight space is expressed as
(17)
wb  [ wb
wb ]  [cb  sb cb  sb ]
where w and w are the lower and upper bounds of w;
c and s indicate the center and spread of w, respectively.
Output Space: The output of the T2FCMAC is the algebraic sum of the receptive-field space and weight
space. The output i(t) of the T2FCMAC can be expressed as [27]
n
n
1 n n

(18)
i t     h b wb   hb wb  .
j 1  1
2  b1  1

Note that the output () of T2FCMAC can be obtained
directly without using the so-called Karnik-Mendel algorithm. Thus, the computation cost of T2FCMAC can be
reduced. This leads the T2FCMAC to be more practical.
As above description, the total adjustable parameters
1
2
3
of the T2FCMAC are p1 , p 2 , p 3 , p , p , p ,  , c,
B

B
415
(20)
xl t  1  xl t   vl t  1
where w, c1, c2 are constant parameters, and rand1l, rand2l
are random variables in [0, 1]. In order to make sure the
components of separated signals Y(t) are low dependence each other, we choose second-order correlation
measure D to be the fitness function (or objective function for minimization) for PSO algorithm.
xl t  1
Gbest t 
vl t  1
vl t 
Pbest t 
xl t 
Figure 5. Behavior illustration of particle swarm optimization.

HDi t 
HDi1 t  1
y li t 
Dil t  1
HDil t  1
…
…
…
…
…
y ip t 
…
…
…
…
…
 ip t 
Di1 t  1
y1i t 
 il t 
…
B. Particle swarm optimization for interval type-2 fuzzy
CMAC’s optimization
To enhance the performance of the BSS with
T2FCMAC-based learning rates, the T2FCMAC system
is optimized by the particle swarm optimization (PSO)
approach. PSO is an evolutionary algorithm based on
population [18, 21, 24, 26]. It is motivated by social behavior of organisms such as bird flicking and fish
schooling. The population-based searching procedure in
which individuals called particles change their position
with time. The particles will fly around in a multidimensional search space and each particle adjusts its position
according to its own experience. The fitness function
evaluates each solution to decide whether it will contribute to the next sample time of solutions. The behavior
illustration of PSO algorithm is shown in Fig. 5. Each
particle moves to a new position xl(t+1) according to the
new velocity vl(t+1) which includes its previous velocity
vl(t), and the moving vectors according to the past best
solution Pbest l t  and global best solution Gbest t  .
The particle position updates laws are
vl t  1  w  vl t   c1  rand 1l  Pbest i t   xl t 
(19)
 c2  rand 2 l  Gbesti t   xl t 
i1 t 
…
and s. As above description, we implement the
T2FCMAC with Di(t) and HDi(t) two inputs to find the
suitable learning rate i(t) for the BSS.
Di t 
Dip t  1
HDip t  1
Figure 6. PSO for optimizing the T2FCMAC for the BSS system.
The proposed optimization method to adjust the
T2FCMAC parameters by PSO for the BSS system is
shown in Fig. 6. Herein, the particle denotes the adjustable parameters of the T2FCMAC for providing the
proper learning rate. Thus, consider the T2FCMAC having n layers and nB blocks, the partial representation has
nnB 27, for fuzzification space, and nnB2, for
weight space, parameters. As shown in Fig. 6, the population size is p and the objective of optimization is to
find the proper T2FCMAC’s parameters to minimize the
value of Di(t). The second-order correlation measure D
and the high order correlation measure HD will also be
computed in each time instant. Assume Di(t) and HDi(t)
computed in the first sample time are two inputs of the
T2FCMAC to compute the Dl(t+1) and HDl(t+1) for
each particle. According to fitness value Dl(t+1), the past
best (Pbest) and global best (Gbest) solutions can be
identified. And the global best solution of Dl(t+1) and
HDl(t+1) are the inputs of next sample time optimization.
As the description of the proposed approach, Fig. 7
summaries the proposed T2FCMAC_PSO approach for
BSS problem.
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
416
Start
Fns=E[D]
Initial
Input S
Fns<Fns_PB
X=AS
Y
Pbest particle
is updated
N
Dependence
measure
Y=WX
Fns<Fns_GB
Y
Gbest particle
is updated
P
S
O
o
p
ti
m
iz
at
io
n
 0.2421
0.4554
W 0   
0.4652

0.8892
0.2694 0.0795 0.4915 
0.1249 0.6701 0.2824 
.
0.2364 0.5248 0.5241

0.2570 0.6839 0.9248 
(23)
The EASI algorithm with fixed learning rate (denoted
EASI),
fuzzy-based learning rate (denoted EASI-Fuzzy)
T2FCMAC l
Particle
l
[35],
fuzzy-based
learning rate with PSO optimization
η
position update
(denoted
EASI-Fuzzy_PSO),
Type-1 FCMAC-based
EASI
l
adaptive
learning
rate
adjustment
with PSO optimization
Y
l=l+1
l
l
Dependence D , HD
(denoted T1FCMAC_PSO), T2FCMAC-based learning
measure
DD
N HD  HD
rate adjustment without PSO optimization (denoted
Y
l<=p
Iter  Iter  1
T2FCMAC), and T2FCMAC-based adaptive learning
rate adjustment with PSO optimization (denoted
l=1
Y Iter<=IterMax
N
End
T2FCMAC_PSO) are introduced to have the comparisons for illustrating the performance of our approach.
Figure 7. The flowchart of the proposed T2FCMAC_PSO for
The fixed learning rate of the traditional EASI is chosen
BSS problem.
as =0.2. For EASI_Fuzzy, the fuzzy inference system is
developed for selecting the proper learning rate for the
EASI algorithm [35]. The D and HD are used for antecedent part input variables, and the consequent part variable is the learning rate  for the EASI algorithm. Five
linguistic variables are set for antecedent part, D {Sml 1,
Sml 2, Sml 3, Mid, Big} and HD {Sml 1, Sml 2, Sml 3,
Mid, Big}, and nine linguistic variables for consequent
part are {Sml 1, Sml 2, Sml 3, Mid 1, Mid 2, Mid 3, Big
1, Big 2, Big 3}. The corresponding fuzzy inference system with 25 fuzzy rules is shown in Table 1. The corresponding membership functions of antecedent and conFigure 8. The source signals.
sequent parts for (D, HD) and  are shown in Fig. 9(a)
and
9(b), respectively. Note that the membership func4. Simulation Results
tions of  is given by the fuzzy singleton as {0.05, 0.1,
In order to demonstrate the effectiveness of the pro- 0.12, 0.15, 0.18, 0.2, 0.22, 0.26, 0.31}. Furthermore, we
posed adaptive learning rate determined by compare with the fuzzy-based learning rate approach,
T2FCMAC_PSO, we consider the following four source adjusted by PSO (EASI-Fuzzy_PSO), for BSS. The
fuzzy-based learning rate system is developed the same
signals as follows [21]
as above description, but the consequent weighting vec sin 2 25t sin2 800t  
tor of the system for providing proper learning rates  is
sin2 300t  6 cos2 60t 
.
(21) optimized by PSO algorithm. The fuzzy inference rules
St   


noiset 
and the membership functions of D and HD in the ante





sign
cos
2
155
t

cedent are the same as Table 1 and Fig. 9, respectively,


The source signals are an amplitude-modulated signal, a but the membership function of the consequent, is iniphase-modulated signal, a square-wave signal, and a tialized in the range [0, 1], which is optimized by PSO
noise signal uniformly distributed in [-1, 1], respectively, algorithm. Since there are nine fuzzy term sets for conshown in Fig. 8. The entries of 4×4 mixing matrix A are sequent part, the particle representation has nine (9) parandom numbers in the range [-1, 1]. In order to compare rameters. The particle number of EASI-Fuzzy_PSO is
the performances with other methods for learning rate double number of the particle representation (18). The
selection, the mixing matrix A and separating matrix are parameters in PSO are given by w=0.3, c1=0.8, and
c2=0.8, respectively.
chosen randomly as follows
For the T2FCMAC, the initial parameters ( p1 , p 2 ,
 0.9183 0.2357 0.8770 0.6966 
1
2
3
0.8209 0.6027 0.3678 0.2367 
p 3 , p , p , p ,  ), are selected randomly in the own


(22)
A
0.9516 0.5406 0.8168 0.1356 
block and satisfies the restricted conditions and


0.5    1 for the interval type-2 triangular asymmetric
0.8295 0.7794 0.7102 0.2952
D, HD
GbInd  l
N
GbInd
GbInd
s1
1
0
-1
0
0.05
0.1
0.15
Time
0.2
0.25
0.3
0.05
0.1
0.15
Time
0.2
0.25
0.3
0.05
0.1
0.15
Time
0.2
0.25
0.3
0.05
0.1
0.15
Time
0.2
0.25
0.3
s2
1
0
-1
0
s3
1
0
-1
0
s4
1
0
-1
0


M.-T. Huang et al.: Type-2 Fuzzy Cerebellar Model Articulation Controller-Based Learning Rate Adjustment
Table 1. Fuzzy inference rules for learning rates adjustment.
D
HD
Sml 1
Sml 2
Sml 3
Mid
Big
Sml 1
Sml 2
Sml 3
Mid
Big
Sml 1
Sml 1
Sml 2
Mid 1
Mid 3
Sml 1
Sml 2
Sml 2
Mid 2
Big 1
Sml 2
Sml 2
Sml 3
Mid 3
Big 2
Mid 1
Mid 2
Mid 3
Big 2
Big 3
Mid 3
Big 1
Big 2
Big 3
Big 3
1.2
Degree of membership
1
Sml 1
Sml 2
Sml 3
Mid
Big
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
D/HD
0.6
0.7
0.8
0.9
1
(a)
1.2
Degree of membership
1
0.8
0.6
0.4
0.2
0
Sml 1
Sml 2
Sml 3
Mid 1
Mid 2

Mid 3
Big 1
Big 2
Big 3
(b)
Figure 9. The membership functions of antecedent and consequent parts for (D, HD) and  in EASI-Fuzzy, (a) antecedent
part for (D, HD); (b) consequent part for and .
20
EASI
EASI - Fuzzy
EASI - Fuzzy_PSO
EASI -T2FCMAC_PSO
EASI - T2FCMAC
EASI - T1FCMAC_PSO
18
16
14
PI
12
10
8
6
4
2
0
0
500
1000
1500
Sample
2000
2500
Figure 10. Comparison results of PI.
3000
417
fuzzy membership function. The parameters c and s
in the weight space are initialized in the range of [0, 1].
Since n=4 and nB=3, the particle has 192 parameters.
The population size is set to be half number of parameter
(96). In PSO, the parameters are given by w=0.35,
c1=0.75, and c2=0.75, respectively.
The total sample time for the BSS problem is 3000
and the comparison results of performance index PI are
shown in Fig. 10. We can observe that the proposed
T2FCMAC_PSO performs the better results than other
methods. Even though the fuzzy-based method
(EASI-Fuzzy) shows the better performance than the
fixed one, the EASI-Fuzzy_PSO, EASI-T1FCMAC, and
the T2FCMAC_PSO methods are more flexible since the
weights or membership functions are adjusted by PSO
optimization. Compare the performance of the fuzzy
system, type-1 FCMAC, and T2 FCMAC from Fig. 10,
we have found that the T2FCMAC perform better than
the T1FCMAC and traditional fuzzy system. Furthermore, the novel T2FCMAC system presents faster speed
of convergence.
From the results of comparisons in Fig. 10, we can
conclude that the EASI learning rate adjustment methods
EASI-Fuzzy and EASI-T2FCMAC without PSO optimizing processing are valid for the BSS problem if the
corresponding fuzzy membership functions and rule base
are designed properly. For the practical application, the
T2FCMAC with using PSO (brown-line in Fig. 10) performs 3.3 seconds in 3000 data. The average processing
of T2FCMAC without PSO method is 1.1ms. This illustrates the efficient of the T2FCMAC approach. In addition, the PSO is used to optimize the systems adaptively
and result better performance than the results without
PSO methods. This illustrates the effectiveness of the
optimization.
The corresponding dependent measures of second-order correlation measure D and high order correlation measure HD for each learning rate selection methods are shown in Fig. 11 (Fig. 11(a) , (b), (c), and (d)
show the dependent measure of separated states for
EASI-Fuzzy method, EASI-Fuzzy_PSO method,
T1FCMAC_PSO, and T2FCMAC_PSO methods, respectively. From Figs. 10 and 11, we can find the dependent measures of separated state are directly proportional to the PI for the BSS. The proposed
T2FCMAC_PSO method performs the faster convergent
speed and better performance than other two methods.
In addition, the variations trajectories of the leaning
rate (t) of the EASI-Fuzzy, EASI-Fuzzy_PSO,
T1FCMAC_PSO, and the T2FCMAC_PSO are shown in
Figs. 12(a), 12(b), 12(c), and 12(d), respectively. The
EASI-Fuzzy_PSO and the T2FCMAC_PSO methods
present the fluctuant learning rate differ from
fuzzy-based method in the beginning of the sample time
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
418
1
HD1
D1
1
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1500 2000
Sample
HD2
D2
2
0.5
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0
0
3000
perform the asymmetric type at final state. In addition,
the separated signals of T2FCMAC-based method are
shown in Fig. 14. Not till 1000 sample time, the signals
are separated steady with the T2FCMAC-based method.
The separated signals are similar as the source signals in
1000 to 3000 sample time. Consequently, the adaptive
leaning rates for the BSS present the fast speed of convergence
than
fixed
one.
Moreover,
the
T2FCMAC_PSO-based BSS perform the fast speed of
convergence and better performance than other methods.
(a)
0.5
0
0
500
1000
D2
500
1000
500
1000
1500 2000
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1500 2000
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1
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0
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1
0.5
0
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1000
0.5
0
0
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HD4
1
500
1
0.5
0
0
0.5
0
0
3000
HD3
D3
2500
1
1
D4
1500 2000
Sample
HD2
2
0
0
5. Conclusions
1
HD1
D1
1
0.5
0
0
3000
(b)
1
HD1
D1
1
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0
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0
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1
HD4
1
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0
0
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0
0
500
1
HD3
1
D3
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0
0
D4
1500
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HD2
D2
1
0.5
2500
0.5
0
0
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(c)
1
HD1
D1
1
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0
0
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1
0.5
0
0
3000
1
0.5
0
0
1000
0.5
0
0
3000
HD4
1
500
1
0.5
0
0
0
0
3000
HD3
1
D3
2500
1
0
0
D4
1500 2000
Sample
HD2
D2
2
0.5
3000
0.5
0
0
(d)
Figure 11. Dependent measure of separated states, (a)
EASI-Fuzzy method; (b) EASI-Fuzzy_PSO method; (c)
T1FCMAC_PSO; (d) T2FCMAC_PSO method.
because they are optimized by PSO. However, the
T2FMAC_PSO method performs the stable learning rate
earlier than EASI-Fuzzy_PSO method.
For the T2FCMAC_PSO method we proposed, the final (after optimization by PSO) interval type-2 triangular
asymmetric fuzzy membership functions for D, HD, and
weighting interval sets are shown in Figs. 13(a) and
13(b), respectively. The triangular membership functions
In this paper, we have proposed the T2FCMAC- based
learning rate adjustment for the BSS and employed the
PSO algorithm to enhance the performance of the
T2FCMAC-based learning rate. In order to illustrate the
performance of the proposed approach, simulation and
comparison results with other methods are introduced.
Simulation results demonstrate that the proposed
T2FCMAC_PSO performs the faster speed of convergence and lower performance.
According to the results, the proposed approach has
the following advantages. (a) Under the EASI algorithm,
the adaptive leaning rate selection methods show the
faster speed convergence than fixed learning rate in the
BSS. (b) With the second-order and higher order correlation coefficients of output components, the learning rate
of EASI algorithm can be determined to balance the
mis-adjustment and the speed of convergence. (c) For
the learning rate selection, the proposed T2FCMAC system not only has better learning ability, but also is more
flexible since the system is optimized by PSO. (d) The
proposed approach without using PSO optimization can
also achieve the signal separation and the T2FCMAC
still performs better than others. (e) In addition, the corresponding fuzzy rules for learning rate adjustment can
be obtained from the T2FCMAC_PSO. (f) The
T2FCMAC can achieve better performance than others
in less fuzzy rules.
Some components in this paper still have some room
for improvement and have expansibility. Therefore, there
are some directions can be further investigated as follows. (a) The performance of the T2FCMAC_PSO adjusted by PSO has associate with the initial state of the
parameters and the computational effort is large. How to
set the initial parameters and make the system more efficient
should
be
discussed.
(b)
The
T2FCMAC_PSO-based BSS method is considered as a
linear system here. However, the nonlinear system is
more realistic for natural world. For the nonlinear system,
the T2FCMAC_PSO-based BSS method may face some
restrictions in the system.

1
M.-T. Huang et al.: Type-2 Fuzzy Cerebellar Model Articulation Controller-Based Learning Rate Adjustment
0.4
1
0.2
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
3
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Layer 3
1
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
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
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1
1.2
1.4
1.6
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1
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0
-0.2
(a)
(b)
2

1
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w23
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w31
w32
w33
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w41
1
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(b)
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w12
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Layer 4
1
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1
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w42
w43

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Sample
2000
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3000
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Sample
2000
2500
3000
500
1000
1500
Sample
2000
2500
3000
(c)
Figure 13. Membership functions after optimization, (a) for D;
(b) for HD; (c) interval sets of consequent part.
0.5

3
0
0
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0
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y1

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y2

4
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(c)
1

1
y3
1
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2500
3000
1
3
2000
0.5
0
0

1500
Sample
y4

2
1

4
2
Figure 14. The separated signals in the BSS with
T2FCMAC_PSO method.
1
0
0
Acknowledgment
(d)
Figure 12. Leaning rate , (a) EASI-Fuzzy; (b)
EASI-Fuzzy_PSO;(c)T1FCMAC_PSO;(d) T2FCMAC_PSO.
Layer 1
1
0.5
0
0.2
0.4
0.6
0.8
Layer 2
1
1
1.2
1.4
1.6
0.5
0
0
0.5
1
Layer 3
1
The authors would like to thank anonymous reviewers
and committee members of Special Issue of iFUZZY
2013 for their insightful comments and valuable suggestions. This work was supported in part by the National
Science Council, Taiwan, R.O.C., under contracts
NSC-102-2221-E-005-095-MY2, NSC-102-2221-E-005061-MY3 and NSC-102-2218-E-005-012.
1.5
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0.5
0
0.2
0.4
1
0.6
Layer 4
0.8
1
1.2
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Meng-Tzu Huang was born in Taiwan,
R.O.C., in 1989. She received the M. S.
degree in Electrical Engineering in 2013
in Yuan Ze University, Taiwan. Her research interests include fuzzy neural
systems, signal processing, adaptive
control, image processing, and blind
source separation systems.
Ching-Hung Lee is currently a Professor of the Department of Mechanical
Engineering at National Chung Hsing
University. Dr. Lee received the Wu
Ta-Yu Medal and Young Researcher
Award in 2008 from the National Science Council, R.O.C. He also received
the 2009 Youth Automatic Control Engineering Award from Chinese Automatic Control Society. In additional, he is an IEEE Senior
Member. His main research interests are fuzzy neural systems,
fuzzy logic control, neural network, signal processing, nonlinear control systems, image processing, and robotics control.
Chih-Min Lin is currently a Chair Professor and the Dean of the College of
Electrical and Communication Engineering, Yuan Ze University, Taiwan. He
is also an Honorary Professor of Obuda
University, Hungary. His research interests include fuzzy system, neural network, cerebellar model articulation controller, automatic control system, and
robotics. He has published 152 journal papers and 155 conference papers. He is an Associate Editor of IEEE Transaction
on Cybernetics, Asian Journal of Control, International Journal
of Fuzzy Systems, and International Journal of Machine
Leaning and Cybernetics. In addition, he is an IEEE Fellow
and IET Fellow; he also serves as a Member of Board of Governors of IEEE Systems, Man, and Cybernetics Society.